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Pseudo-Fermi surface and phonon softening in sodium with a stepwise electron distribution

The absorption of light by a metal disturbs the electron distribution around the Fermi surface. Here, we calculate the phonon dispersion relations of free-electron-like metal, bcc sodium, with a stepwise electron distribution function by using a model pseudo-potential method. The step can behave as a pseudo-Fermi surface, which produces the singularities at specific wavenumbers in the response function. The singularity gives rise to long-range oscillations in the interatomic potential and results in imaginary phonon frequencies around the N point.